We consider Kirchhoff equations with a small parameter ${\varepsilon}$ such as $$ \varepsilon {u_{{\varepsilon}}}''(t)+(1+t)^{-p}{u_{{\varepsilon}}}'(t)+ \m{{u_{{\varepsilon}}}(t)}A{u_{{\varepsilon}}}(t)=0. $$ We prove the existence of global solutions when $\varepsilon$ is small with respect to the size of initial data, for all $0\leq p \leq 1$ and $\gamma \geq 1$. Then we provide global-in-time error estimates on ${u_{{\varepsilon}}} - u$ where $u$ is the solution of the parabolic problem obtained setting formally $\varepsilon = 0$ in the previous equation.

We consider an initial-boundary value problem for the hyperelastic rod wave equation (a special case of which is the Camassa-Holm equation). We establish the existence of a global-in-time weak solution. Moreover, we prove a ``weak equals strong'' uniqueness result.

In this work we investigate the existence of global weak solutions and asymptotic behavior for a contact problem associated with a nonlinear, generalized thermoelastic coupled system.

We study the global well-posedness of the nonlinear wave equation $$ u_{tt} - \Delta u - \Delta _p u_t = f(u) $$ in a bounded domain ${\Omega} \subset \mathbb{R}^n$ with Dirichlét boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent; i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H^1_0({\Omega})$ into $L^2({\Omega})$. The nonlinear term $- \Delta _p u_t $ is a strong damping where the $-\Delta _p$ denotes the p-Laplacian (defined below). Under suitable assumptions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of a local weak solution. Also, such a unique solution depends continuously on the initial data from the finite energy space. In addition, we prove that weak solutions are global, provided the exponent of the damping term dominates the exponent of the source.

We prove a local higher integrability result for the gradient of solutions to singular, parabolic equations of $p$-Laplacian type. To this end, we show that solutions satisfy a reverse H\"older inequality on intrinsic cylinders, whose geometry depends on the $L^r$-norm of the solution. The exponent $r \geq 2$ allows us to derive estimates in the subcritical range $1 < p \leq 2N/(N+2)$.

We see that the generalized Fourier transform due to A.G. Ramm for the case of $n=3$ space dimensions remains valid, with some modifications, for all space dimensions $n\ge 2$. We use the resulting spectral representation of the exterior Laplacian to study exterior problems. In particular the Fourier splitting method developed by M.E. Schonbek extends easily to the study of this type of problems, as we illustrate for the dissipative 2-dimensional quasi-geostrophic equation in the critical case.